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Mathematical Communications 4(1999), 269-275 269

International mathematical olympiad

Zeljko Hanjs

Today mathematical competitions are very popular with primary and secondaryschool students and there are many countries all around the world where they areregularly organised. There are several rounds and a lot of students are included,especially at the beginning rounds. The best students from the previous roundhave the right to continue on the higher level of competition. The final level for thesecondary school student competitors is the International Mathematical Olympiad

(IMO). The team for the IMO from Croatia is determined at the National Compe-tition which is held in May.

The first mathematical competitions were organised in Hungary in 1894, and inRomania in 1898. Mathematical competitions in Croatia for the secondary schoolstudents started in 1959 and next year the first Federal Competition was held, whichwas then organised every year until 1991.

Romania was the initiator of the first international competition. The idea oforganizing it came from the Romanian mathematician Tiberiu Roman in 1956,and mathematics is still his great love, although he is 83 years old. After detailedpreparations the first International Mathematical Olympiad was held in Romania in1959, as well as the second one in 1960. At the beginning only the following countriesfrom the Eastern Europe participated: Bulgaria, East Germany, Czechoslovakia,Hungary, Poland, Romania and the USSR. In 1963 Yugoslavia participated for thefirst time, and after that new and new countries from Europe arrived. The firstOlympiad in Yugoslavia took place in Cetinje, Montenegro, in 1967, and the secondone in Belgrade in 1977. 21 countries took part at that 19 th IMO. Cuba was thefirst non-European country which participated at the 13th IMO, in 1971, and itwas the host country in 1987. Australia participated for the first time at the 22 ndIMO in 1981, and was the host country in 1988, when the 200th anniversary ofEuropeans inhabiting that continent was. In 1980 the IMO was not organized,and only some local olympiads were held. In 1993 Croatia as well as Bosnia andHerzegovina, Macedonia and Slovenia became regular members of IMO.

In 1999 the 40 th IMO was organized in Romania, and it was the fifth one heldin this country (the previous ones had been held in 1959, 1960, 1969, 1978). Duringthe last few years there where about 80 countries and 450 contestants at the IMO.

The lecture presented at the Mathematical Colloquium in Osijek organized by Croatian

Mathematical Society - Division Osijek, May 28, 1999.Department of Mathematics, University of Zagreb, Bijenicka 30, HR-10 000 Zagreb, Croatia,

e-mail: hanjs@math.hr

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270 Z. Hanjs

A few words from

the Regulations of the International Mathematical Olympiad

The aims of the IMO are: to discover, encourage, and challenge mathematically gifted young people in

all countries; to foster friendly relations between mathematicians of all countries; to create opportunities for the exchange of the information on school syllabuses

and practice throughout the world.

Participation and Responsibility

The contestants should not have been formally enrolled at a university or anyequivalent post-secondary institution and on the first day of contest they shouldnot be older than 20 years.

The Organizers of the IMO cover all official expenses for Leaders, Deputy Lead-ers and Student Contestants, including meals and accommodation, for the periodof the official program. The Organizers cover the costs of all contest activities.

Deputy Leaders are responsible for the conduct of their students during thewhole period of the IMO.

Proposals for Problems

Each participating country is invited to submit up to six proposed problems,with solutions, to the Problem Selection Committee, until the deadline given by theOrganizers.

Proposed problems should cover various fields of the pre-university mathematicsand should be of varying degrees of difficulty.

The Problem Selection Committee will prepare a short list of at least 25 problemsand no more than 30 problems with their solutions, for consideration by the Jury.

Jury Regulations

The International Jury shall consist of all Leaders of participating countries anda Chairman appointed by the IMO Organizing Committee.

The Jury should confirm that the total number of prizes will not exceed one halfof the number of the Student Contestants; that the established number of prizeswill be distributed as the first, second and third prizes in a ratio approximating1:2:3 as closely as possible.

A certificate of honorable mention shall be awarded to each Student Contestantwho does not receive a prize and who has obtained full marks on at least onequestion.

Special prizes shall be awarded for a complete solution of outstanding meritonly.

The Jury approves the translations of the chosen problems into all requiredlanguages.

Contest Regulations

Each of two contest periods lasted four and a half hours.The only instruments permitted during the contest are writing and mechanical

drawing instruments.

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During the first half hour of each examination period each Student Contestantmay submit, on a specially provided note a paper, a written question for considera-tion by the Jury.

The Coordinators together with the Leader and the Deputy Leader of eachcountry decide on the scores during a coordination session.

The Leader of the country which submitted the problem, will for each of thesix problems, respectively, verify the coordination of the solution given by StudentContestants from the host country.

A breaf overlook on the IMO

The official beginning of the IMO is the first Jury meeting and three days later isthe official arrival of student contestants with their Deputy Leaders. Every partici-pating country has its representative in the Jury. It has to prepare the problems for

the competition. The problems have been divided in four groups: algebra, geome-try, number theory and combinatorics. Six problems have to be chosen, three forthe first day and three for the second day. The first problem should be the easiest,the second one more difficult than the first and the third, the most difficult of thefirst day problems. The first problem of the second day should be the easiest one ofthe second day, the second one should be more difficult than this one, and the thirdshould be most difficult of all problems. This one is obtaining for the special prize,and it is very hard to achieve it. For the first of all the English version is prepared,and after that the versions in official languages: French, Chinese, German, Russianand Spanish. After that the Leader of every country translates the problems intothe official language of his country. Each contestant may obtain a version in hisown language and in one of the official ones.

The day before the first day of contest the Opening Ceremony of the IMO takesplace. At that moment the Leaders have to be strictly separated from the StudentContestants and the Deputy Leaders. On the first day of contest after answeringthe student questions an excursion for the members of the Jury is organized. Afterthe second day of contest excursions are prepared for the students, they can spendtheir time on computers and sport, but they also have a possibility to keep companywith students from other countries.

Two days after the contest Leaders and Deputy Leaders have to coordinatethe solutions of the students. At the end, according to the results of the contest,the Jury decides about the prizes. An honorable mention was introduced at theOlympiad in Australia.

Up to 1991 the Student Contestants from Croatia, as members of the Yugosla-vian team, obtained 7 silver and 10 bronze medals, and one honorable mention.

From 1993 to 1999 they obtained 2 silver and 18 bronze medals as well as 4 honor-able mentions. Our Contestants, who have been three times at the IMO are: MladenBestvina from Osijek (1976 1978), Miroslav Jurisic from Split (1993 1995), An-drijana Radovcic from Split (1997 1999) and Matija Kazalicki from Zagreb (1997 1999). Silver medals were won by: Damir Henc from Zagreb (1969), Mladen Bestv-ina (silver + bronze + silver), Pavle Padzic from Zagreb (1982), Miroslav Jurisicfrom Kastel (1990), Bojan Antolovic from Zagreb (1996) and Vedran Zoric from

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Split (1997). The first Contestant from Croatia was Valerijan Bjelik from Vinkovci(1963).

A great majority of Student Contestants decide to study mathematics, and manyof them are offered a job at the Department of Mathematics in Zagreb, or in someother faculties. Some of them go to study abroad and return home after some years,but some of them continue to work in some foreign country.

In Croatia a great care is dedicated to the preparation of student contestants aswell as organising the Winter and Summer Mathematical Meetings.

Now there is a plan of organizing the IMO for 8 years in future: South Korea(2000), SAD (2001), Great Britain (2002), Japan (2003), Greece (2004), Iran (2005),Slovenia (2006) and Vietnam (2007).

Members of the Croatian Olympiad team at the 40 th IMO, from the left:Andrijana Radovcic, Tomislav Pejkovic, Srdan Maksimovic, Mislav Miskovic,Miodrag Cristian Iovanov (Team leader in Romania), Marinko Jablan, Matija

Kazalicki, te Ilko Brnetic and Zeljko Hanjs (Team leaders)

Logo of the 40th IMO

Every IMO has its Logo, and some of them are very interesting. Let us have a

look at the Logo for the last IMO which was held in Romania.

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Logo for the 40th IMO

The Logo for IMO 99 was inspired by an old and very popular elementary prob-lem in Romanian high schools. It is called the five lei coin problem. The Roma-nian curency is called leu and its plural form is lei. The English translationof leu is lion. So, this logo makes reference to the coin of five lei (lions). Thelegend says that, using this coin, the famous Romanian geometer Gheorghe Titeicacreated during the joke the following problem: three equal circles having a commonpoint, pairwisely intersect again at three points which lie on a fourth circle equalto the previous ones.

In the given logo, the given circles are colored red, yellow and blue, which arethe colors of the Romanian national flag. The forth circle is painted black and itis used to create number 40, the rank of this IMO, while the letter O from IMOhelps us to obtain five circles in all, the total number of circles used in the OlympicGames.

The author of this Logo is Professor Bogdan Enescu, a former golden medallist

of the 20th IMO. Please, have some fun solving the five lei coin problem!

Some other International Mathematical Competitions

Mediterranean Mathematical Competition

Austrian-Polish Mathematical Competition

Polish-Israel Mathematical Competition

Ibero-American Mathematical Competition

Asian-Pacific Mathematical Competition The Tournament of the Towns

Some other information about the International Mathematical Olympiads can befound on Internet, at:

http://www.math.hr/hmd

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274 Z. Hanjs

The problems from the 40th IMO

First day, 16 July 1999

Problem 1. Determine all sets S of at least three points in the plane whichsatisfy the following condition:

for any two distinct points A and B in S, the perpendicular bisector of line segmentAB is an axis of symmetry for S.

Problem 2. Let n be a fixed integer, with b 2.

1. Determine the least constant C such that the inequality

1i

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International mathematical olympiad 275

Literatura

[1] Mathematics Competitions, Journal of the WFNMC, Australia.

[2] Matematicke olimpijade (Mathematical Olympiads), Element, Zagreb, 1997.

[3] Z. Matjaz, Altius, Citius, Fortius, The Society of Mathematics, Physics andAstronomy of Slovenia, Ljubljana, 1995.

[4] 40. National Math Olympiads in Slovenia, The Society of Mathematics, Physicsand Astronomy of Slovenia, Ljubljana, 1995.

[5] IX & XIX International Mathematical Olympiads, Mathematicians Society ofSerbia, Beograd, 1997.

[6] International and Balkan Mathematical Olympiads, 1984-1995, Mathemati-cians Society of Serbia, Beograd, 1996.

[7] Problems of the Austrian-Polish Mathematical Competition, 1978-1993, TheAcademic Distribution Center, Freeland, Maryland, 1994.

[8] USA Mathematical Olympiads 1972-1986, The Mathematical Association ofAmerica, 1988.

[9] The Canadian Mathematical Olympiad 1969-1993, Canadian Mathematical So-ciety, Toronto, 1993.

[10] A. Gardner, The Mathematical Olympiad Handbook An Introduction toProblem Solving, Oxford University Press, Oxford-New York-Tokyo, 1997.

[11] A. Engel, Problem-Solving Strategies, Springer-Verlag New York, Inc., 1998.

[12] Matematicko-fizicki list, Zagreb.

[13] Mathematics and Informatics Quarterly, International journal for mathematicsfor secondary school students.

[14] Editions of Elementary Mathematics, Mathematical Competitions, Element,Zagreb.